Theorem expival 9478

Description: Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)

Assertion

Ref	Expression
expival	⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))

Proof of Theorem expival

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iftrue 3356	. . . . 5 ⊢ (𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) = 1)
2		ax-1cn 7069	. . . . 5 ⊢ 1 ∈ ℂ
3	1, 2	syl6eqel 2169	. . . 4 ⊢ (𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ)
4	3	a1i 9	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ))
5		elnnz 8361	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
6		elnnuz 8655	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1))
7	5, 6	bitr3i 184	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 0 < 𝑁) ↔ 𝑁 ∈ (ℤ≥‘1))
8	7	biimpi 118	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 0 < 𝑁) → 𝑁 ∈ (ℤ≥‘1))
9	8	adantll 459	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → 𝑁 ∈ (ℤ≥‘1))
10		cnex 7097	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
11	10	a1i 9	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → ℂ ∈ V)
12		simpl 107	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ≥‘1)) → 𝐴 ∈ ℂ)
13		elnnuz 8655	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ ↔ 𝑧 ∈ (ℤ≥‘1))
14		fvconst2g 5396	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℕ) → ((ℕ × {𝐴})‘𝑧) = 𝐴)
15	13, 14	sylan2br 282	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ≥‘1)) → ((ℕ × {𝐴})‘𝑧) = 𝐴)
16	15	eleq1d 2147	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ≥‘1)) → (((ℕ × {𝐴})‘𝑧) ∈ ℂ ↔ 𝐴 ∈ ℂ))
17	12, 16	mpbird 165	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
18	17	adantlr 460	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 𝑧 ∈ (ℤ≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
19	18	adantlr 460	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) ∧ 𝑧 ∈ (ℤ≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
20		mulcl 7100	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ)
21	20	adantl 271	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
22	9, 11, 19, 21	iseqcl 9443	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁) ∈ ℂ)
23		iftrue 3356	. . . . . . . . . . . 12 ⊢ (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) = (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁))
24	23	eleq1d 2147	. . . . . . . . . . 11 ⊢ (0 < 𝑁 → (if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ ↔ (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁) ∈ ℂ))
25	24	adantl 271	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → (if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ ↔ (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁) ∈ ℂ))
26	22, 25	mpbird 165	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ)
27	26	ex 113	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
28	27	adantr 270	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
29	28	3adantl3 1096	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
30		simpll2 978	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℤ)
31	30	znegcld 8471	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℤ)
32		simpr 108	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 < 𝑁)
33	30	zred 8469	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℝ)
34		0red 7120	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 ∈ ℝ)
35	33, 34	lenltd 7227	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 ≤ 0 ↔ ¬ 0 < 𝑁))
36	32, 35	mpbird 165	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ≤ 0)
37		simplr 496	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 𝑁 = 0)
38	37	neneqad 2324	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ≠ 0)
39	38	necomd 2331	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 ≠ 𝑁)
40		0z 8362	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℤ
41		zltlen 8426	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
42	40, 41	mpan2 415	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
43	42	3ad2ant2 960	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
44	43	ad2antrr 471	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
45	36, 39, 44	mpbir2and 885	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 < 0)
46	33	lt0neg1d 7616	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ 0 < -𝑁))
47	45, 46	mpbid 145	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 < -𝑁)
48		elnnz 8361	. . . . . . . . . . . 12 ⊢ (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
49	31, 47, 48	sylanbrc 408	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℕ)
50		elnnuz 8655	. . . . . . . . . . 11 ⊢ (-𝑁 ∈ ℕ ↔ -𝑁 ∈ (ℤ≥‘1))
51	49, 50	sylib 120	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ (ℤ≥‘1))
52	10	a1i 9	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ℂ ∈ V)
53	17	3ad2antl1 1100	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ 𝑧 ∈ (ℤ≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
54	53	adantlr 460	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 𝑧 ∈ (ℤ≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
55	54	adantlr 460	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) ∧ 𝑧 ∈ (ℤ≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
56	20	adantl 271	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
57	51, 52, 55, 56	iseqcl 9443	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) ∈ ℂ)
58		simpll1 977	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐴 ∈ ℂ)
59		expivallem 9477	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ -𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0)
60	59	3com23 1144	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ -𝑁 ∈ ℕ ∧ 𝐴 # 0) → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0)
61	60	3expia 1140	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ -𝑁 ∈ ℕ) → (𝐴 # 0 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0))
62	58, 49, 61	syl2anc 403	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐴 # 0 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0))
63	39	neneqd 2266	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 = 𝑁)
64		ioran 701	. . . . . . . . . . . . 13 ⊢ (¬ (0 < 𝑁 ∨ 0 = 𝑁) ↔ (¬ 0 < 𝑁 ∧ ¬ 0 = 𝑁))
65	32, 63, 64	sylanbrc 408	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ (0 < 𝑁 ∨ 0 = 𝑁))
66		zleloe 8398	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
67	40, 66	mpan 414	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
68	67	3ad2ant2 960	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
69	68	ad2antrr 471	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
70	65, 69	mtbird 630	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 ≤ 𝑁)
71	70	pm2.21d 581	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (0 ≤ 𝑁 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0))
72		simpll3 979	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐴 # 0 ∨ 0 ≤ 𝑁))
73	62, 71, 72	mpjaod 670	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0)
74	57, 73	recclapd 7869	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)) ∈ ℂ)
75		iffalse 3359	. . . . . . . . . 10 ⊢ (¬ 0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) = (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))
76	75	eleq1d 2147	. . . . . . . . 9 ⊢ (¬ 0 < 𝑁 → (if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ ↔ (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)) ∈ ℂ))
77	76	adantl 271	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ ↔ (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)) ∈ ℂ))
78	74, 77	mpbird 165	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ)
79	78	ex 113	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → (¬ 0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
80		zdclt 8425	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 0 < 𝑁)
81	40, 80	mpan 414	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → DECID 0 < 𝑁)
82		df-dc 776	. . . . . . . . . 10 ⊢ (DECID 0 < 𝑁 ↔ (0 < 𝑁 ∨ ¬ 0 < 𝑁))
83	81, 82	sylib 120	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
84	83	adantl 271	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
85	84	adantr 270	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
86	85	3adantl3 1096	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
87	29, 79, 86	mpjaod 670	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ)
88		iffalse 3359	. . . . . . 7 ⊢ (¬ 𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) = if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))))
89	88	eleq1d 2147	. . . . . 6 ⊢ (¬ 𝑁 = 0 → (if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ ↔ if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
90	89	adantl 271	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → (if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ ↔ if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
91	87, 90	mpbird 165	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ)
92	91	ex 113	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (¬ 𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ))
93		zdceq 8423	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
94	40, 93	mpan2 415	. . . . 5 ⊢ (𝑁 ∈ ℤ → DECID 𝑁 = 0)
95		df-dc 776	. . . . 5 ⊢ (DECID 𝑁 = 0 ↔ (𝑁 = 0 ∨ ¬ 𝑁 = 0))
96	94, 95	sylib 120	. . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
97	96	3ad2ant2 960	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
98	4, 92, 97	mpjaod 670	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ)
99		sneq 3409	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
100	99	xpeq2d 4387	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (ℕ × {𝑥}) = (ℕ × {𝐴}))
101		iseqeq3 9436	. . . . . . 7 ⊢ ((ℕ × {𝑥}) = (ℕ × {𝐴}) → seq1( · , (ℕ × {𝑥}), ℂ) = seq1( · , (ℕ × {𝐴}), ℂ))
102	100, 101	syl 14	. . . . . 6 ⊢ (𝑥 = 𝐴 → seq1( · , (ℕ × {𝑥}), ℂ) = seq1( · , (ℕ × {𝐴}), ℂ))
103	102	fveq1d 5200	. . . . 5 ⊢ (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦))
104	102	fveq1d 5200	. . . . . 6 ⊢ (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦))
105	104	oveq2d 5548	. . . . 5 ⊢ (𝑥 = 𝐴 → (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦)))
106	103, 105	ifeq12d 3368	. . . 4 ⊢ (𝑥 = 𝐴 → if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦))) = if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦))))
107	106	ifeq2d 3367	. . 3 ⊢ (𝑥 = 𝐴 → if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)))) = if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦)))))
108		eqeq1 2087	. . . 4 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
109		breq2 3789	. . . . 5 ⊢ (𝑦 = 𝑁 → (0 < 𝑦 ↔ 0 < 𝑁))
110		fveq2 5198	. . . . 5 ⊢ (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁))
111		negeq 7301	. . . . . . 7 ⊢ (𝑦 = 𝑁 → -𝑦 = -𝑁)
112	111	fveq2d 5202	. . . . . 6 ⊢ (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))
113	112	oveq2d 5548	. . . . 5 ⊢ (𝑦 = 𝑁 → (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))
114	109, 110, 113	ifbieq12d 3375	. . . 4 ⊢ (𝑦 = 𝑁 → if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦))) = if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))))
115	108, 114	ifbieq2d 3373	. . 3 ⊢ (𝑦 = 𝑁 → if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦)))) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))
116		df-iexp 9476	. . 3 ⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)))))
117	107, 115, 116	ovmpt2g 5655	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))
118	98, 117	syld3an3 1214	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661  DECID wdc 775   ∧ w3a 919   = wceq 1284   ∈ wcel 1433   ≠ wne 2245  Vcvv 2601  ifcif 3351  {csn 3398   class class class wbr 3785   × cxp 4361  ‘cfv 4922  (class class class)co 5532  ℂcc 6979  0cc0 6981  1c1 6982   · cmul 6986   < clt 7153   ≤ cle 7154  -cneg 7280   # cap 7681   / cdiv 7760  ℕcn 8039  ℤcz 8351  ℤ_≥cuz 8619  seqcseq 9431  ↑cexp 9475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432  df-iexp 9476

This theorem is referenced by:  expinnval  9479  exp0  9480  expnegap0  9484